Reaction Rates - How to Determine the Instantaneous Rate of Reaction

The instantaneous rate of reaction in chemistry refers to the rate at a specific moment in time, often at a particular point along the reaction's progress. It's the rate of change in the concentration of a reactant or product at a specific time, and it can be determined using calculus and the concept of the derivative. To find the instantaneous rate of reaction, you typically follow these steps:
Choose the specific time (t) at which you want to calculate the instantaneous rate. This will be your time of interest.
Measure the concentration of the reactant or product at that specific time. This will be your concentration at the chosen time, denoted as [A]t or [B]t, where A or B represents the reactant or product.
Choose a very small time interval around the chosen time (t). This small time interval is often denoted as Δt and is essentially a very short period of time during which you assume the rate of reaction to be approximately constant.
Measure the concentrations of the reactant or product at two different times: one just before the chosen time (t) and one just after (t + Δt). These concentrations are denoted as [A]t-Δt and [A]t+Δt (or [B]t-Δt and [B]t+Δt).
Use the following formula to calculate the instantaneous rate of reaction:
Instantaneous Rate = -d[A]/dt or d[B]/dt
Where d[A]/dt represents the rate of change of concentration with respect to time for the reactant A, and d[B]/dt represents the rate of change of concentration with respect to time for the product B.
Mathematically, you can approximate this rate by calculating the slope of the concentration-time graph at the specific time (t) you're interested in. If you have a function representing the concentration of the reactant or product as a function of time, you can differentiate it with respect to time and then substitute the chosen time (t) into the derivative expression.
Remember that the instantaneous rate of reaction can vary over time as the reaction progresses, so you may want to calculate it at different time points to get a complete picture of how the rate changes throughout the reaction.
In summary, to determine the instantaneous rate of reaction, choose a specific time of interest, measure the concentrations at that time and around it, and use calculus to calculate the derivative of concentration with respect to time at that specific moment.